Weighted Fréchet spaces of holomorphic functions

Transkrypt

STUDIA MATHEMATICA 174 (3) (2006)
Weighted Fréchet spaces of holomorphic functions
by
Elke Wolf (Paderborn)
Abstract. This article deals with weighted Fréchet spaces of holomorphic functions
which are defined as countable intersections of weighted Banach spaces of type H ∞ . We
characterize when these Fréchet spaces are Schwartz, Montel or reflexive. The quasinormability is also analyzed. In the latter case more restrictive assumptions are needed to obtain
a full characterization.
I. Introduction. This article deals with the weighted Fréchet spaces
HW (G) and HW0 (G) of holomorphic functions. For an increasing sequence
W = (wn )n∈N of strictly positive continuous functions (weights) on an
open subset G of CN we consider the projective limit of the Banach spaces
Hwn (G) := {f ∈ H(G) ; kf kn := supz∈G wn (z)|f (z)| < ∞} resp. H(wn )0 (G)
:= {f ∈ H(G) ; wn f vanishes at ∞ on G}, n ∈ N. Under rather general
assumptions we give a characterization of being Schwartz, Montel and reflexive in terms of the sequence of weights (which are considered as growth
conditions in the sense of [13]) or in terms of their associated growth conditions. Using the class W of weights on the unit disk which was introduced
by Bierstedt and Bonet [9] we get a necessary and sufficient condition for
the quasinormability of HW (G) resp. HW0 (G).
In the case of Köthe echelon spaces these characterizations were obtained
by Köthe, Grothendieck, Dieudonné–Gomes, Bierstedt–Meise–Summers,
and Valdivia. For spaces of continuous functions the characterizations in
terms of the weights were obtained by Bierstedt, Meise and Summers (see
[14] and [15]). In this paper these questions are studied in the setting of
weighted spaces of holomorphic functions. In contrast to the case of continuous functions the so-called associated growth conditions mentioned by
Andersen and Duncan in [1] and studied thoroughly by Bierstedt, Bonet and
Taskinen in [13] are needed to get the characterizations. While we get results
2000 Mathematics Subject Classification: Primary 46E10; Secondary 46A04, 46A11,
46A13, 46A25.
Key words and phrases: weighted Fréchet space of holomorphic functions, Schwartz,
Montel, reflexive, quasinormable.
[255]
256
E. Wolf
for being Schwartz, Montel and reflexive under rather mild assumptions, the
characterization of the quasinormability of HW (G) resp. HW0 (G) requires
the restriction to the unit disk D as well as to the class W of radial weights
which was introduced by Bierstedt and Bonet (see [9]). This class provides
a suitable frame to decompose holomorphic functions in a certain sense.
This article is organized as follows. Section II gives the necessary notation and definitions. Section III studies weighted Fréchet spaces of holomorphic functions which are reflexive resp. Montel. We show that the algebraic
identity HW (G) = HW0 (G) is equivalent to the reflexivity of HW0 (G)
if W = (wn )n∈N is an increasing sequence of strictly positive continuous
functions on an open set G ⊂ CN , N ≥ 1. Moreover we characterize in
terms of weights considered as growth conditions and associated growth
conditions when HW (G) is a Montel space. Here we use methods of Bierstedt and Bonet (see [11]). We finish this section with some remarks about
the Schwartz property of weighted Fréchet spaces of holomorphic functions.
A result of Bonet, Friz and Jordá [19] yields a characterization in terms of
the associated growth conditions under rather general conditions.
Section IV consists of two parts. The first part presents a necessary
condition for the quasinormability of HW (G) resp. HW0 (G). In the second
part we prove that this condition is also sufficient when we restrict our
attention to the class W of radial weights on the unit disc. Under some
additional assumptions we show that HW0 (G) is quasinormable if and only
if it satisfies condition (QNo) of Peris.
Acknowledgments. This article arises from a part of the author’s doctoral thesis which was supervised by Klaus D. Bierstedt and José Bonet. The
author thanks both of them for all their advice, helpful remarks and valuable
suggestions. The discussions with them have been particularly stimulating.
II. Notation. Our notation for locally convex spaces is standard; see for
example Jarchow [26], Köthe [27], Meise and Vogt [31] and Pérez Carreras
and Bonet [32]. For a locally convex space E, we denote by E ∗ the space of
all linear functionals on E while E ′ is the topological dual and Eb′ the strong
dual. If E is a locally convex space, U0 (E) and B(E) stand for the families of
all absolutely convex 0-neighborhoods and absolutely convex bounded sets
in E, respectively.
In what follows, G denotes an open subset of CN , N ≥ 1. The space H(G)
of all holomorphic functions on G will usually be endowed with the topology
co of uniform convergence on compact subsets of G. Let W = (wn )n∈N be
an increasing sequence of strictly positive continuous functions on G. For
every n ∈ N the spaces
Hwn (G) := {f ∈ H(G) ; kf kn := sup wn (z)|f (z)| < ∞},
z∈G
257
H(wn )0 (G) := {f ∈ H(G) ; wn f vanishes at ∞ on G}
endowed with the norm k · kn are Banach spaces. The weighted Fréchet
spaces of holomorphic functions are defined by
HW (G) := projn Hwn (G),
HW0 (G) := projn H(wn )0 (G).
For each n ∈ N, let Bn , resp. Bn,0 , be the closed unit ball of Hwn (G), resp.
H(wn )0 (G), and Cn := Bn ∩ HW (G), resp. Cn,0 := Bn,0 ∩ HW0 (G). By
B n , B n,0 , C n , C n,0 we denote the co-closures
of the corresponding sets.
The sequence n1 Cn n∈N , resp. n1 Cn,0 n∈N , constitutes a 0-neighborhood
base of HW (G), resp. HW0 (G). Without loss of generality we may assume
that (Cn )n∈N , resp. (Cn,0 )n∈N , is a 0-neighborhood base. Put
W := {w : G → ]0, ∞[ ;
w continuous on G, wn w is bounded on G for all n ∈ N},
and Cw := {f ∈ HW (G) ; |f | ≤ w on G}, Cw,0 := Cw ∩ HW0 (G), w ∈
W . We write C w and C w,0 for the respective co-closures. Then (Cw )w∈W ,
resp. (Cw,0 )w∈W , is a fundamental system of bounded subsets of HW (G),
resp. HW0 (G). Each Cw is absolutely convex and co-compact. (See [7, Section 3.A].)
Let v be a weight on G considered as a growth condition in the sense
of [13]. Its associated growth condition is defined by
ve(z) := sup{|g(z)| ; g ∈ H(G), |g| ≤ v},
z ∈ G.
A weight v on a balanced domain G ⊂ CN , N ≥ 1, is said to be radial if
v(z) = v(λz) for every λ ∈ C such that |λ| = 1. On a balanced open subset
G of CN , N ≥ 1, each f ∈ H(G) has a Taylor series representation about
zero,
∞
X
f (z) =
pk (z), z ∈ G,
k=0
where pk is a k-homogeneous polynomial (k = 0, 1, . . . ). The series converges
to f uniformly on each compact subset of G. The Cesàro means of the partial
sums of the Taylor series of f are denoted by Sn (f ) (n = 0, 1, . . . ); that is,
[Sn (f )](z) =
1 XX
pk (z) ,
n+1
n
l
l=0
k=0
z ∈ G.
Each Sn (f ) is a polynomial (of degree ≤ n) and Sn (f ) → f uniformly on
every compact subset of G (f ∈ H(G) arbitrary).
Bierstedt, Bonet and Galbis (see [12]) used Cesàro means to show that
if W = (wn )n∈N is an increasing sequence of non-negative continuous and
radial functions on a balanced open set G ⊂ CN , N ≥ 1, such that HW0 (G)
contains the polynomials, then B n,0 = Bn and C n = Bn for every n ∈ N.
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E. Wolf
Weighted Fréchet spaces of continuous functions are defined similarly on
locally compact spaces X. Bierstedt and Meise (see [14]) and Bastin (see
[2], [3]) characterized properties like Schwartz, Montel, quasinormable, distinguished and the density condition for these spaces. They gave conditions
in terms of the weights which, in a modified form, also play a role in this
article.
We now mention some examples of weighted Fréchet spaces of holomorphic functions which were studied in the literature.
(I) In [22] Epifanov considered a bounded convex domain G in CN ,
N ≥ 1, a decreasing sequence (ϕn )n∈N of bounded non-negative convex
functions on G and defined the sequence W = (wn )n∈N by
wn (z) = e−ϕn (z) ,
z ∈ G, n ∈ N.
He showed that under certain assumptions the strong dual of HW (G) is
topologically isomorphic to the inductive limit VH(CN ), where V = (vn )n∈N
is given by
vn (z) = e− sup{Rehz,ti−ϕn (t) ; t∈G} ,
z ∈ CN , N ≥ 1.
For further information on weighted inductive limits of holomorphic functions see e.g. [16] and [6].
(II) Let a plurisubharmonic function p : CN → [0, ∞) satisfy the conditions log(1 + |z|2 ) = O(p(z)), p(z) = p(|z|) and p(2z) = O(p(z)). The
increasing sequence W = (wn )n∈N is defined by
wn (z) := e−p(z)/n ,
z ∈ G, n ∈ N.
The weighted Fréchet space HW (G) was considered by Berenstein, Chang
and Li in [4] and denoted by A0p (G). Note that HW (G) is a Schwartz space.
They studied geometrical conditions on V := f −1 (0), where f ∈ A0p (CN ), in
order that V is an interpolating variety for the corresponding ring. In case
G = C and p(z) = |z|̺ , ̺ > 0, we obtain the Fréchet space of all entire
functions of order ̺ and of type 0.
(III) Weighted Fréchet spaces of holomorphic functions are used by Bonet
and Meise to consider (LF )-spaces in connection with ultradistributions of
Roumieu type (see [20]). These (LF )-spaces appear as the Fourier–Laplace
transform of spaces of ultradistributions of compact support. By [20] the
(LF )-spaces are nuclear and reflexive. The “steps” of the inductive limit
are given by the following Fréchet spaces. Let Ω be a non-empty open and
convex subset of RN with 0 ∈ Ω, and (Kn )n∈N be a fundamental sequence
◦
◦
of convex and compact subsets of Ω such that 0 ∈ K 1 and Kn ⊂ K n+1 for
every n ∈ N. The increasing sequences Wn = (wn,k )k∈N , n ∈ N, are defined
by
1
wn,k (z) := exp −hn (Im(z)) − ω(z) ,
k
259
z ∈ CN , k ∈ N,
where hn is given by hn (x) := supy∈Kn hx, yi, x ∈ RN , n ∈ N, and ω denotes
a non-quasianalytic function. For further information on non-quasianalytic
functions we refer to [20].
III. Montel and reflexive spaces. To get the desired characterization
when HW0 (G) is reflexive we need the following lemma.
Lemma 1. Consider a sequence (fj )j∈N ⊂ HW0 (G). The following are
equivalent:
(a) fj → 0 with respect to σ(HW0 (G), HW0 (G)′ ).
(b) (fj )j∈N is bounded , and (fj )j∈N converges to 0 with respect to co.
The proof is standard (cf. [8], [17], [34]).
Corollary 2. On each bounded subset B of HW0 (G), co is stronger
than the weak topology.
The proof follows directly from Lemma 1.
Theorem 3. Let Cw = C w,0 for every w in a subset A of W such that
(Cw )w∈A is a fundamental system of bounded sets in HW (G). Then the
following assertions are equivalent:
(a) HW (G) = HW0 (G).
(b) HW0 (G) is reflexive.
Proof. (b)⇒(a). We fix f ∈ HW (G). There is w ∈ A with f ∈ Cw . Since
Cw = C w,0 , we can find a sequence (fj )j∈N ⊂ Cw,0 ⊂ HW0 (G) with fj → f
with respect to co; thus, (fj )j∈N is relatively weakly compact by (b). We
can find a subsequence (fjk )k∈N of (fj )j∈N which converges weakly to an
element g of HW0 (G), hence pointwise to g. We conclude g = f ∈ HW0 (G)
and finally HW (G) = HW0 (G).
(a)⇒(b). We assume HW (G) = HW0 (G). By definition we have to show
that each bounded subset B of HW0 (G) is relatively weakly compact. There
is w ∈ A such that B ⊂ Cw,0 . Since HW (G) = HW0 (G) we have Cw = Cw,0 ,
and Cw,0 is co-compact. By Corollary 2 and Montel’s theorem B is weakly
relatively compact.
For a sequence W = (wn )n∈N with wn = wn+1 = w for every n ∈ N
we obtain a Banach space Hw(G). For spaces of this type the following is
known:
Theorem 4 (Bonet and Wolf [21]). Let G be an open subset of CN ,
N ≥ 1, and let v be a strictly positive and continuous weight on G. Then
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E. Wolf
the space Hv0 (G) embeds almost isometrically into c0 . In particular , if this
space is infinite-dimensional , then Hv0 (G) and Hv(G) are not reflexive.
There are many examples of sequences W = (wn )n∈N such that Cw =
C w,0 for every w ∈ A. See [12] and [24].
Remark 5 (Bierstedt, Bonet and Galbis [12, Proposition 1.2(c)]). Let
W = (wn )n∈N be an increasing sequence of strictly positive continuous and
radial functions on a balanced open set G ⊂ CN , N ≥ 1. The Cw for w ∈ W
radial form a basis of bounded sets in HW (G) and Cw = C w ∩ HW0 (G) =
C w,0 for every w ∈ W radial.
Proposition 6 (Holtmanns [24, Proposition 4.2.8]). Let G = {z ∈ C ;
Im(z) > 0} and W = (wn )n∈N be an increasing sequence of strictly positive
continuous functions on G such that limIm(z)→0 wn (z) = 0 and wn (z) ≤
wn (z + ip) for every 0 < p < 1 and every n ∈ N. Then Cw = C w,0 for every
w ∈ W.
In what follows we first study under which conditions HW (G) is Montel
and then the connection between Montel spaces and reflexive spaces. We
need the following lemma.
Lemma 7. The following assertions are equivalent:
(a) HW (G) is Montel.
(b) The topology τ of HW (G) and co coincide on each bounded subset
of HW (G).
Proof. (a)⇒(b). Let HW (G) be a Montel space. By definition each
bounded subset B of HW (G) is τ -relatively compact. By [27, 3.2.(6)], τ coincides with each weaker topology, so in particular with co. Hence (b) follows.
(b)⇒(a). We fix a bounded subset A of HW (G). Then there is w ∈ W
such that A ⊂ Cw and Cw is co-compact. Hence A is co-relatively compact
and by (b) relatively compact in HW (G). By definition this shows that
HW (G) is a Montel space.
Theorem 8. The following assertions are equivalent:
(a) HW (G) = HW0 (G), and HW (G) is a Montel space.
(b) For every n ∈ N and every w ∈ W , wn (w)∼ vanishes at ∞ on G.
Proof. (a)⇒(b). We assume that HW (G) is a Montel space and that
HW (G) = HW0 (G). We fix n ∈ N and put A := {wn (z)δz ; z ∈ G}. Since
A is contained in Cn◦ , A is equicontinuous in HW (G)′ . It follows that
σ(HW (G)′ , HW (G))|A = λ(HW (G)′ , HW (G))|A ,
where λ(HW (G)′ , HW (G)) denotes the topology of uniform convergence on
compact subsets of HW (G).
261
For fixed U ∈ U(HW (G)′ , σ(HW (G)′ , HW (G))) there is a finite set
(f1 , . . . , fs ) ⊂ HW (G) = HW0 (G) such that
A ∩ (f1 , . . . , fs )◦ ⊂ U ∩ A.
We write V := (f1 , . . . , fs )◦ and because of HW (G) = HW0 (G) we can find
K ⊂⊂ G with
wn (z)δz ∈ V
for every z ∈ G \ K.
We fix w ∈ W and ε > 0. Since HW (G) is Montel, Cw is compact in HW (G).
Hence for the λ(HW (G)′ , HW (G))-0-neighborhood εCw◦ ⊂ HW (G)′ we can
find K ⊂⊂ G such that wn (z)δz ∈ εCw◦ for every z ∈ G \ K, i.e. there exists
K ⊂⊂ G such that
ε ≥ wn (z) sup{|f (z)| ; f ∈ Cw } = wn (z)(w)∼ (z)
for every z ∈ G \ K. So (b) follows.
(b)⇒(a). In order to show HW (G) = HW0 (G), we fix f ∈ HW (G). We
can find w ∈ W with |f | ≤ (w)∼ on G. For fixed n ∈ N we obtain
wn |f | ≤ wn (w)∼
on G.
This together with assertion (b) implies that wn |f | vanishes at ∞ on G.
Hence f belongs to HW0 (G) and HW (G) = HW0 (G) as desired.
It remains to show that HW (G) is Montel. We fix w ∈ W and claim that
HW (G) and (H(G), co) induce the same topology on Cw . Since (Cw )w∈W is
a fundamental system of bounded subsets of HW (G), Lemma 7 then implies
that HW (G) is a Montel space. We fix n ∈ N and ε > 0. By assumption
there is K ⊂⊂ G such that
wn (z)(w)∼ (z) ≤ ε
for every z ∈ G \ K. We put
U := {g ∈ HW (G) ; wn (z)|g(z)| ≤ ε for all z ∈ G}
and have to show that there is a 0-neighborhood V in (H(G), co) with
V ∩ Cw ⊂ U ∩ Cw .
We put V := {g ∈ H(G) ; |g(z)| ≤ ε/maxy∈K wn (y) for all z ∈ K} and fix
g ∈ V ∩ Cw . Thus, we obtain wn (z)|g(z)| ≤ maxy∈K wn (y)ε/maxy∈K wn (y)
= ε for z ∈ K. For z ∈ G \ K we distinguish two cases. If (w)∼ (z) > 0, we
get
|g(z)|
≤ ε.
wn (z)|g(z)| = wn (z)(w)∼ (z)
(w)∼ (z)
If we have (w)∼ (z) = 0, this implies g(z) = 0 because g belongs to Cw . Then
wn (z)|g(z)| = 0 < ε. Thus g is in U ∩ Cw .
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E. Wolf
Corollary 9. Let Cw = C w,0 for every w ∈ A ⊂ W such that (Cw )w∈A
is a fundamental system of bounded sets in HW (G). Then the following
assertions are equivalent:
(a) HW (G) is Montel.
(b) For every n ∈ N and every w ∈ W , wn (w)∼ vanishes at ∞ on G.
Proposition 10. Let G be an open and connected subset of C such
that C∗ \ G has no one-point component. Then HW (G) = HW0 (G) implies
condition (b) in Theorem 8.
Proof. Assume that condition (b) is not satisfied. There are w ∈ W ,
n ∈ N and a sequence (zk )k∈N ⊂ G which converges to z0 in the boundary
of G in C∗ such that for every k ∈ N we have
wn (zk )(w)∼ (zk ) ≥ ε.
(1)
By [13, Proposition 1.2(iv)] for every k ∈ N there is fk ∈ H(G) such that
|fk | ≤ w on G and |fk (zk )| = (w)∼ (zk ). Each fk is an element of HW (G).
Without loss of generality we may assume that z0 is contained in a
closed, connected set L ⊂ C∗ \ G of more than one point. It is easy to see
that U := C∗ \ L is conformally equivalent to D.
The sequence (zk )k∈N contains an H ∞ (U )-interpolating subsequence
which will also be denoted by (zk )k∈N . (See [23, p. 195].)
Then the proof of [35, III.E.4] yields the existence P
of a sequence of functions (ϕj )j∈N ⊂ H ∞ (U ) such that ϕj (zk ) = δjk and ∞
j=1 |ϕj | ≤ M < ∞.
P∞
P∞
Put f :=
ϕ
f
.
We
have
|f
(z)|
≤
|ϕ
(z)f
k
k (z)| ≤ M w(z) for
k=1 k k
k=1
every z ∈ G. Thus the series converges uniformly on compact subsets of G,
hence f ∈ H(G).
It remains to show that f belongs to HW (G) \ HW0 (G). First we prove
that f is an element of HW (G). For every z ∈ G and every m ∈ N, we have
wm (z)|f (z)| ≤ wm (z)
∞
X
|ϕk (z)| |fk (z)| ≤ wm (z)M w(z) < ∞.
k=1
P
∼
On the other hand, |f (zj )| = | ∞
k=1 ϕk (zj )fk (zj )| = |fj (zj )| = (w) (zj ) for
∼
each j ∈ N. An application of (1) shows wn (zk )|f (zk )| = wn (zk )(w) (zk ) ≥ ε
for every n ∈ N. So f 6∈ H(wn )0 (G). This contradicts HW (G) = HW0 (G).
Corollary 11. Let G ⊂ C be open and connected such that C∗ \ G has
no one-point component. Assume that Cw = C w,0 for every w ∈ A. Then
the following assertions are equivalent:
(a)
(b)
(c)
(d)
HW (G) = HW0 (G).
HW0 (G) is reflexive.
HW (G) is reflexive.
For every n ∈ N and every w ∈ W , wn (w)∼ vanishes at ∞ on G.
263
(e) HW (G) is Montel.
(f) HW0 (G) is Montel.
Proof. The equivalence of (a) and (b) is given by Theorem 3. By Theorem 8 and Proposition 10 we deduce that (a) is equivalent to (d). Since
HW0 (G) is a closed topological subspace of HW (G), (c) implies (b) by [26,
Proposition 11.4.5] and (e) yields (f) by [26, Proposition 11.5.4]. By Theorem 8, (e) follows from (d). Since HW (G) and HW0 (G) are Fréchet spaces
we obtain (e)⇒(c) and (f)⇒(b).
Remark 12. The following condition (M ′ ) was introduced by Bierstedt,
Meise and Summers in [16] for Köthe matrices and studied for sequences of
weights by Bierstedt and Meise in [14].
An increasing sequence W = (wn )n∈N of strictly positive continuous
functions on an open set G ⊂ CN , N ≥ 1, is said to satisfy condition (M ′ ) if
for any n ∈ N and any subset Y of G which is not relatively compact there
is m = m(n, Y ) > n such that inf y∈Y wn (y)/wm (y) = 0.
They showed that the algebraic equality CW (G) = CW0 (G) is equivalent
to (M ′ ) (see [14, Proposition 5.5]). It is an easy consequence that (M ′ )
implies HW (G) = HW0 (G). A simple computation shows that (M ′ ) is
satisfied if and only if the following holds: For every n ∈ N and every w ∈ W ,
wn w vanishes at ∞ on G.
This remark shows that (b) in Theorem 8 differs from (M ′ ) exactly by
the use of an associated growth condition. We will show that this is really
a difference by constructing an example in which HW (G) = HW0 (G), but
CW (G) 6= CW0 (G). The idea for the construction of this example is taken
from [13]. First we need some auxiliary results.
The following condition (N D ′ ) was introduced by Bierstedt and Meise
in [14]. It is well known that a sequence W of weights which has (N D ′ ) does
not satisfy (M ′ ).
Remark 13. Let W = (wn )n∈N be an increasing sequence of strictly
positive continuous functions on an open set G ⊂ CN , N ≥ 1. Then W is
said to satisfy condition (N D ′ ) if there are n ∈ N and a decreasing sequence
(Jk )k∈N such that for every k ≥ n we have
(i) inf z∈Jk wn (z)/wk (z) > 0,
(ii) there is l(k) > k with inf z∈Jk wn (z)/wl(k) (z) = 0.
Another condition which is needed is the following condition (S ′ ) for an
increasing sequence W = (wn )n∈N of strictly positive continuous functions
on an open set G ⊂ CN , N ≥ 1, which was studied by Bierstedt, Meise and
Summers in [15] in the setting of Köthe sequence spaces:
(S ′ )
For every n ∈ N there is m > n such that wn /wm vanishes at ∞ on G.
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E. Wolf
It is well known that (S ′ ) implies (M ′ ). But the converse is not true
(see [15]).
Lemma 14. Let G1 be an open subset of C, and put G := G1 × C. Let
T = (tn )n∈N and U = (un )n∈N be increasing sequences of weights on G1
and C, respectively. Let 0 < p < 1, and assume that for each n ∈ N and
each z ∈ C, 1 ≤ un (z) ≤ (1 + |z|)p . Choose S = (sn )n∈N with sn (z) :=
un (z)/(1 + |z|)p , z ∈ C, and put W = (wn )n∈N , wn (z1 , z2 ) = tn (z1 )sn (z2 )
for (z1 , z2 ) ∈ G, n ∈ N. Then:
(a) Every f ∈ HW (G) is constant in the second coordinate z2 , i.e. we
have f (z1 , z2 ) = f (z1 , z2′ ) for z2 , z2′ ∈ C, z1 ∈ G1 .
(b) HW (G) and HT (G1 ) are canonically isomorphic.
Proof. (a) We fix f ∈ HW (G). By definition there is M > 0 such that
wn (z1 , z2 )|f (z1 , z2 )| = tn (z1 )sn (z2 )|f (z1 , z2 )| ≤ M for every (z1 , z2 ) ∈ G. By
assumption we have 1/(1 + |z2 |)p ≤ sn (z2 ) ≤ 1 for every z2 ∈ C, hence we
obtain
M
M
1
|f (z1 , z2 )| ≤
≤
(1 + |z2 |)p on G.
tn (z1 ) sn (z2 )
tn (z1 )
By Liouville’s theorem the claim follows (see [13, Lemma 2.2].
(b) Consider the maps
ψ : HW (G) → HT (G1 ),
ψ : HT (G1 ) → HW (G),
−1
f 7→ (z1 7→ f (z1 , 0))
g 7→ ((z1 , z2 ) 7→ g(z1 )),
which obviously are topological isomorphisms.
Example 15. In Lemma 14 consider G1 = C and choose an increasing
sequence T = (tn )n∈N of weights tn on C such that tn (0) = 1 for every n ∈ N
and such that T has (S ′ ). Hence HT (C) = HT0 (C). For every increasing
sequence S = (sn )n∈N on C with lim|z|→∞ sn (z) = 0 for every n ∈ N and
such that S satisfies the assumptions of Lemma 14 we define W = (wn )n∈N
on C2 = C × C as in Lemma 14. We claim HW (C2 ) = HW0 (C2 ). To show
this we have to distinguish two cases.
Case 1: We suppose |z1 | → ∞ and fix f ∈ HW (C2 ) as well as n ∈ N.
By Lemma 14 we have HW (C2 ) ∼
= HT (C). Since lim|z2 |→∞ sn (z2 ) = 0 there
is M > 0 with |sn (z)| ≤ M for every z ∈ C. Fix ε > 0 and put ε′ := ε/M .
Since HT (C) = HT0 (C), we can find K1 ⊂⊂ C such that, by the proof of
Lemma 14,
wn (z1 , z2 )|f (z1 , z2 )| = tn (z1 )sn (z2 )|f (z1 , 0)| ≤ M tn (z1 )|f (z1 , 0)| ≤ ε′ M = ε
for every z1 ∈ C \ K1 and every z2 ∈ C. Now, take K = K1 × {0} to get
wn (z1 , z2 )|f (z1 , z2 )| ≤ ε for every z = (z1 , z2 ) ∈ C2 \K. Hence f ∈ HW0 (C2 ).
265
Case 2: We assume |z2 | → ∞ and fix f ∈ HW (C2 ) as well as n ∈ N. By
Lemma 14 we have HW (C2 ) ∼
= HT (C) = HT0 (C). Hence there is m > 0
with
wn (z1 , z2 )|f (z1 , z2 )| = tn (z1 )|f (z1 , 0)|sn (z2 ) ≤ msn (z2 )
for every z = (z1 , z2 ) ∈ C2 . Now we fix ε > 0 and put ε′ := ε/m. Since
lim|z2 |→∞ sn (z2 ) = 0 we can find K2 ⊂⊂ C such that
wn (z1 , z2 )|f (z1 , z2 )| ≤ msn (z2 ) ≤ mε
for all z1 ∈ C and z2 ∈ C \ K2 .
Take K = {0} × K2 to get wn (z1 , z2 )|f (z1 , z2 )| ≤ ε for every z = (z1 , z2 ) ∈
C2 \ K. Thus, f ∈ HW0 (C2 ).
Next we construct a sequence S = (sn )n∈N as above such that in addition
W = (wn )n∈N has (N D ′ ), hence not (M ′ ). Therefore CW (G) 6= CW0 (G)
(see [14, Proposition 5.5]).
For n ∈ N, n ≥ 3, and arbitrary k ∈ N0 let first un : R+ → R+ be given
on [2k, 2k + 2] by
1
for r ∈ [2k, 2k + 2−n ] and r ∈ [2k + 1, 2k + 2],
un (r) :=
(1 + r)(n−1)/2n for r ∈ [2k + 3 · 2−(n+1) , 2k + 1 − 2−(n+1) ],
with un affine on [2k + 2−n , 2k + 3 · 2−(n+1) ] and [2k + 1 − 2−(n+1) , 2k + 1].
Distinguishing several cases it is elementary to verify that (un )n≥3 is an increasing sequence. Now extend un radially, un (z) = un (|z|) for z ∈ C, n ∈ N,
to obtain an increasing sequence (un )n∈N on C for which the assumption of
Lemma 14 holds with p = 1/2.
At this point it suffices to show that W satisfies (N D ′ ). For arbitrary
n ∈ N, take Jn = {(0, 2k + 2−m ) ; k, m ≥ n} ⊂ C2 . Given n ≥ n0 := 3, for
any (0, 2k + 2−m ) ∈ Jn we have
1
,
(1 + 2k + 2−m )1/2
1
wn (0, 2k + 2−m ) = tn (0)sn (2k + 2−m ) =
(1 + 2k + 2−m )1/2
wn0 (0, 2k + 2−m ) = tn0 (0)sn0 (2k + 2−m ) =
since m ≥ n; hence inf Jn wn0 /wn = 1 > 0. But for ln := n + 1, k ≥ n,
wn+1 (0, 2k + 2−n ) = tn+1 (0)sn+1 (2k + 2−n ) =
(1 + 2k + 2−n )n/(2n+2)
,
(1 + 2k + 2−n )1/2
wn0 (0, 2k + 2−n )
= (1 + 2k + 2−n )−n/(2n+2).
wln (0, 2k + 2−n )
We obtain inf Jn wn0 /wln ≤ (1 + 2k + 2−n )−n/(2n+2) → 0 (k → ∞), which
proves that W satisfies (N D ′ ).
Now we are interested in the question when HW (G) is Schwartz.
266
E. Wolf
Proposition 16. Let C n = Bn as well as Bn = B n,0 for every n ∈ N.
The following are equivalent:
(a) HW (G) is a Schwartz space.
(b) For every n ∈ N there is m > n such that Hwm (G) ֒→ Hwn (G) is
compact.
(c) For every n ∈ N there is m > n such that H(wm )0 (G) ֒→ H(wn )0 (G)
is compact.
(d) For every n ∈ N there is m > n such that for every ε > 0 there is
K ⊂⊂ G with
wn (z) < εw
em (z)
for every z ∈ G \ K.
Proof. The equivalence of (b), (c) and (d) follows from [19, Theorem 8].
(b)⇒(a). This is [25, Proposition 3.15.9].
(a)⇒(b). If HW (G) is a Schwartz space, for every n ∈ N there is m > n
such that for every ε > 0 we can find a finite set F ⊂ HW (G) with Cm ⊂
F + εCn . This implies
Cm ⊂ F + εCn ⊂ F + εBn .
Since εBn is co-compact and F is co-closed, from [25, Proposition 2.10.5] it
follows that F + εBn is co-closed. We get Bm = C m ⊂ F + εBn and hence
(b) follows.
IV. The quasinormability of HW (G) resp. HW0 (G). First we give
a necessary condition for the quasinormability of HW (G) resp. HW0 (G).
Proposition 17. Let C n = Bn for every n ∈ N and HW (G) be quasinormable. Then the following holds: For every n ∈ N there is m > n such
that for every α > 0 we can find w ∈ W with
1 ∼
α
(2)
on G.
≤w+
wm
wn
Proof. By definition, for every n ∈ N there is m > n such that for every
α > 0 there exists w ∈ W with Cm ⊂ Cw + αCn ⊂ Cw + αBn . Since Cw is
co-compact and αBn is co-closed, Cw + αBn is co-closed and we obtain
(3)
C m = Bm ⊂ Cw + αBn .
It is enough to show that (3) implies (2). For this we fix f ∈ H(G) with
|f | ≤ 1/wm on G. Then f is contained in Bm and hence in Cw + αBn by (3).
Thus f = g1 + g2 with g1 ∈ Cw and g2 ∈ αBn . We get |f | ≤ |g1 | + |g2 |
≤ w + α/wn on G. If we take the supremum over all f , we obtain (2).
Proposition 18. Inequality (2) implies the following condition: For every n ∈ N there is m > n such that for every k > n and every µ > 0 there
is ξ > 0 with
(4)
1
wm
∼
≤
µ
ξ
+
wk
wn
267
on G.
Proof. First, we fix n ∈ N, select m > n as in (2) and fix k > n as well
as µ > 0. Applying (2) we obtain (1/wm )∼ ≤ w + µ/wn on G. By definition
we can find ξ > 0 such that w ≤ ξ/wk on G. This implies (1/wm )∼ ≤
w + µ/wn ≤ ξ/wk + µ/wn on G.
Proposition 19. Let Cw = C w,0 for every w ∈ A ⊂ W such that
(Cw )w∈A is a fundamental system of bounded sets in HW (G). The following
are equivalent:
(a) HW (G) is quasinormable.
(b) HW0 (G) is quasinormable.
Proof. By [7, Proposition 10], HW (G) and (HW0 (G)′b )′b are topologically equal. A Fréchet space E is quasinormable if and only if the strong
bidual E ′′ is quasinormable (see for example [10, Theorem 3]).
The following construction was introduced by Bierstedt and Bonet in [9].
Let W be a class of strictly positive continuous radial weights v on the unit
disc D which satisfy limr→1− v(r) = 0 and for which the restriction of v to
[0, 1) is non-increasing. We suppose that the class W is stable under finite
minima and under multiplication by positive scalars.
Next, we assume that there is a sequence Rn : H(D) → H(D), n ∈ N,
of linear operators which are continuous for the compact open topology
and such that the range of each Rn is a finite-dimensional subspace of the
polynomials. It is also assumed that Rn Rm = Rmin(n,m) for any n, m with
n 6= m and that for each polynomial p there is n such that Rn p = p, from
which it follows that Rm p = p for each m ≥ n. Moreover, we suppose that
there is c > 0 such that sup|z|=r |Rn p(z)| ≤ c sup|z|=r |p(z)| for all n, all
r ∈ (0, 1) and any polynomial p.
Finally, setting R0 := 0, and putting rn := 1 − 2−n , n ∈ N ∪ {0}, we
assume that the following conditions are satisfied by the class W:
(P1)
There is C ≥ 1 such that, for each v ∈ W and for each polynomial p,
1
sup( sup |(Rn+2 − Rn−1 )p(z)|)v(rn)
C n |z|=rn
≤ kpkv ≤ C sup( sup |(Rn+1 − Rn )p(z)|)v(rn ).
n
(P2)
|z|=rn
For each v ∈ W there is D(v) ≥ 1 such that for each sequence
(pn )n∈N of polynomials of which only finitely many are non-zero,
268
E. Wolf
∞
X
sup (Rn+1 − Rn )pn (z)v(z) ≤ D(v) sup( sup |pk (z)|)v(rk ).
z∈D n=1
k
|z|=rk
The main example for W will be the set of all strictly positive continuous
radial weights v on D which satisfy limr→1− v(r) = 0, are non-increasing on
[0, 1), and for which there are ε0 > 0 and k(0) ∈ N such that:
(L1)
(L2)
inf k v(rk+1 )/v(rk ) ≥ ε0 ,
lim supk→∞ v(rk+k(0) )/v(rk ) < 1 − ε0 .
In this case, Rn is the convolution with the P
de la Vallée Poussin kernel, i.e.
k
for a holomorphic function f on D, f (z) = ∞
k=0 ak z , we have
n
Rn f (z) :=
2
X
k=0
k
ak z +
n+1
2X
k=2n +1
2n+1 − k
ak z k .
2n
Conditions (L1) and (L2) are a uniform version of the conditions introduced by W. Lusky in [28], [29], and they also appear in the sequence space
representations for weighted (LB)-spaces given by Mattila, Saksman and
Taskinen [30]. Bierstedt and Bonet showed that (L1) and (L2) imply (P1)
and (P2) (see [9]).
The following lemma is well known. Nevertheless, we give the proof.
Lemma 20. Let E be a locally convex space and F a dense subspace of E.
Assume that F is quasinormable. Then E also is quasinormable.
Proof. We fix a 0-neighborhood base U in F . Since F is dense in E, a
0-neighborhood base of E is given by U := {U ; U ∈ U }. By [26, Proposition
10.7.1] the quasinormability of F implies that for every U ∈ U there is
V ⊂ U , V ∈ U, such that for every λ > 0 there is a bounded set B ⊂ F
with V ⊂ B + λU . We obtain V ⊂ B + λU ⊂ B + 2λU . Since B is bounded,
E is quasinormable.
The converse of this lemma is not true. In [18] Bonet and Dierolf constructed examples of reflexive and quasinormable Fréchet spaces such that
there is a non-distinguished dense subspace.
Theorem 21. Let W = (wn )n∈N ⊂ W be an increasing sequence of
weights on D such that HW0 (D) contains the polynomials. The following
are equivalent:
(a) HW0 (D) is quasinormable.
(b) HW (D) is quasinormable.
(c) For every n ∈ N there is m > n such that for every k > n and every
ε > 0 there is λ > 0 with
Cm,0 ⊂ λCk,0 + εCn,0 .
269
(d) For every n ∈ N there is m > n such that for every k > n and every
µ > 0 there is ξ > 0 with
1 ∼
µ
ξ
+
on D.
≤
wm
wk
wn
(e) For every n ∈ N there is m > n such that for every α > 0 there is
w ∈ W with
α
1 ∼
on D.
≤w+
wm
wn
Proof. (a)⇔(c). This is [31, Definition before Lemma 26.13].
(a)⇔(b). By Remark 5 we have Cw = C w,0 for every w ∈ W radial.
Thus, we can apply Proposition 19 to obtain the desired equivalence.
(a)⇒(e). We have Bn = B n,0 and Bn = C n for every n ∈ N (see [12] as
mentioned above). Hence Proposition 17 implies (d).
(e)⇒(d) See the proof of Proposition 18.
(d)⇒(c). By [12] the polynomials are dense in HW0 (D). Hence it is
enough to consider only polynomials. We fix n ∈ N, choose m > n as
in (d) and fix k > n and ε > 0. Now, we put µ := ε/(2c2 + D2 )2C,
where c and C are the constants from (P1), and the condition before (P1),
and D2 = D(wn ) in (P2). For µ, (d) yields ξ. We fix p ∈ P ∩ Cm,0 .
Hence |p| ≤ 1/wm , or equivalently, |p| ≤ (1/wm )∼ on D. Condition (d)
implies
1 ∼
2ξ 2µ
µ
ξ
+
≤ max
,
≤
on D.
wm
wk
wn
wk wn
Now, we put u := min(wk /2ξ, wn /2µ). Then u belongs to the class W,
and we have
2ξ 2µ
1
1 ∼
,
≤ max
= .
|p| ≤
wm
wk wn
u
Hence u|p| ≤ 1 on D. Put κ1 := 1/2ξ, κ2 := 1/2µ, u1 := wk , u2 := wn , i.e.
u = min(κ1 u1 , κ2 u2 ).
P
P∞
We have p = ∞
n=0 (Rn+1 − Rn )p = R1 p +
n=1 (Rn+1 − Rn )p, and the
sum is finite. We first treat the term R1 p.
By the condition before (P1) and the estimate on u|p|, we get
u(r1 ) sup |R1 p(z)| ≤ cu(r1 ) sup |p(z)| ≤ c.
|z|=r1
|z|=r1
We select i ∈ {1, 2} with u(r1 ) = κi ui (r1 ). From the second inequality
in (P1), applied to the polynomial R1 p and u, and once more the condition
270
E. Wolf
before (P1), we conclude
sup ui (z)|R1 p(z)| ≤ C sup ui (rn )( sup |(Rn+1 − Rn )R1 p(z)|)
n
z∈D
|z|=rn
= Cui (r1 ) sup |(R2 − R1 )R1 p(z)|
|z|=r1
= C(κi )
−1
u(r1 ) sup |(R2 − R1 )R1 p(z)|
|z|=r1
≤ 2cC(κi )
−1
u(r1 ) sup |R1 p(z)| ≤ 2c2 C(κi )−1 .
|z|=r1
4Cc2 ξCk,0
This implies R1 p ∈
or R1 p ∈ P
4Cc2 µCn,0 .
To treat the other term p − R1 p = ∞
n=1 (Rn+1 − Rn )p, we first apply
the first inequality in (P1) for u and the estimate for u|p| to get
(5)
u(rn )( sup |(Rn+2 − Rn−1 )p(z)|) ≤ C.
|z|=rn
We can write N as a disjoint union J1 ∪ J2 such that
κ1 u1 (rj ) for j ∈ J1 ,
u(rj ) =
κ2 u2 (rj ) for j ∈ J2 .
P
Now, for i = 1, 2 put gi =
n∈Ji (Rn+1 − Rn )p, which is a polynomial;
clearly p − R1 p = g1 + g2 . We fix i ∈ {1, 2} and let pin := (Rn+2 − Rn−1 )p
for n ∈ Ji and pin := 0 otherwise. The properties of the sequence (Rn )n∈N
imply
∞
X
X
(Rn+1 − Rn )pin ,
(Rn+1 − Rn )(Rn+2 − Rn−1 )p =
gi =
n=1
n∈Ji
and all the sums are finite; hence
∞
X
(Rn+1 − Rn )pin .
sup ui (z)|gi (z)| = sup ui (z)
z∈D
z∈D
n=1
Since only a finite number of the pin are non-zero and all the weights belong
to the class W, we can apply (P2) and the estimate (5) to conclude
sup ui (z)|gi (z)| ≤ Di sup( sup |pin (z)|)ui (rn ) ≤ Di sup ( sup |pin (z)|)ui (rn )
z∈D
n
|z|=rn
n∈Ji |z|=rn
= Di sup ( sup |(Rn+2 − Rn−1 )p(z)|)ui (rn )
n∈Ji |z|=rn
≤ Di (κi )−1 sup ( sup |(Rn+2 − Rn−1 )p(z)|)u(rn )
n∈Ji |z|=rn
≤ Di (κi )
−1
C.
This yields g1 ∈ 2ξD1 CCk,0 and g2 ∈ 2µD1 CCn,0 . Thus
p = R1 p + g1 + g2 ∈ (2c2 + D1 )2ξCCk,0 + εCn,0 .
Put λ := (2c2 + D1 )2ξC to obtain the claim.
271
The following condition was introduced by Peris in [33].
Definition 22. A locally convex space E is said to satisfy condition
(QNo) if for every U ∈ U there is V ∈ U such that for all ε > 0 we can find
P ∈ L(E, E) with
(i) P (V ) ∈ B(E),
(ii) (id − P )(V ) ∈ εU .
If a Fréchet space E is (QNo) then it is quasinormable (see [33]).
Lemma 23 (Peris [33]). Let E be a locally complete locally convex space
and F a dense subspace with (QNo). Then E also enjoys (QNo).
Proposition 24. Let W = (wn )n∈N ⊂ W be an increasing sequence
of weights on D such that the polynomials P are dense in HW0 (D) and
V := {1/w ; w ∈ W } is a subset of W. The following are equivalent:
(a) HW0 (D) is quasinormable.
(b) HW0 (D) is (QNo).
Proof. (b)⇒(a). This follows from the definition (see [33, remarks after
Definition 3.2]).
(a)⇒(b). This proof is analogous
to that of Theorem 21. Using the same
P
notations we write Ti p := ∞
(R
− Rn )pin for i ∈ {1, 2} and find that
n+1
n=1
T1 p ∈ 2D1 CCw,0 and T2 p ∈ 2λD2 CCn,0 . Thus
p = R1 p + T1 p + T2 p ∈ (2c2 + D1 )2CCw,0 + 2D2 CλCn,0
or
p = R1 p + T1 p + T2 p ∈ 2D1 CCw,0 + (2c2 + D2 )2λCCn,0 .
We distinguish the following cases:
Case 1: We assume u(r1 ) = κ1 u1 (r1 ). We define T : P → P by T (p) =
R1 p + T1 p. Then
T (Cm,0 ∩ P) ⊂ (2c2 + D1 )2C(Cw,0 ∩ P) ∈ B(P),
(I − T )(Cm,0 ∩ P) ⊂ 2D2 Cλ(Cn,0 ∩ P) ⊂ ε(Cn,0 ∩ P).
Case 2: We suppose u(r1 ) = κ2 u2 (r1 ). We define T : P → P by T (p) =
T1 p and get
T (Cm,0 ∩ P) ⊂ 2D1 C(Cw,0 ∩ P) ∈ B(P),
(I − T )(Cm,0 ∩ P) ⊂ (2c2 + D2 )2Cλ(Cn,0 ∩ P) ⊂ ε(Cn,0 ∩ P).
Thus, P and HW0 (D) have (QNo).
We finish this article with an example of a sequence W = (wn )n∈N such
that V belongs to the class W.
272
E. Wolf
Proposition 25. Let W = (wn )n∈N be an increasing sequence of strictly
positive continuous functions on D. Assume that there are ε0 > 0 and k0 ∈ N
such that the following conditions are satisfied :
(L1)
(L2)
inf k wn (rk+1 )/wn (rk ) ≥ ε0 for every n ∈ N.
There is k1 ∈ N with
wn (rk+k0 ) < (1 − ε0 )wn (rk )
for every k ≥ k1 and n ∈ N.
Then V ⊂ W.
Proof. We fix v ∈ W . There is w ∈ W with v ≤ w such that there is
a sequence (βn )n∈N of positive numbers such that for every r > 0 there is
k(r) ∈ N with
1
1
(6)
(1/w)(z) =
wn (z).
= max
min1≤n≤k(r) βn /wn (z) 1≤n≤k(r) βn
We have to show that there are ε2 > 0 and k2 ∈ N such that
(L1)
(L2)
(1/w)(rk+1 )
(1/w)(rk ) ≥ ε2 .
(1/w)(rk+k2 )
lim supk→∞ (1/w)(r
k)
inf k
< 1 − ε2 .
We choose ε2 = ε0 and k2 = k0 and prove (L1). For a fixed k ∈ N it
remains to show
ε2 (1/w)(rk ) ≤ (1/w)(rk+1 ).
We select 0 < rk+1 < s < 1 and by (6) get
(1/w)(z) =
1
wk (z)
1≤n≤k(s) βn
max
for z ∈ D with |z| ≤ s. We distinguish the following cases:
(i) If (1/w)(rk ) = (1/βj )wj (rk ) and (1/w)(rk+1 ) = (1/βj )wj (rk+1 ),
then
1
ε2
ε2 (1/w)(rk ) =
wj (rk ) ≤
wj (rk+1 ) = (1/w)(rk+1 ).
βj
βj
(ii) If (1/w)(rk ) = (1/βj )wj (rk ) and (1/w)(rk+1 ) = (1/βl )wl (rk+1 ), then
ε2
1
1
wj (rk ) ≤
wj (rk+1 ) ≤
wl (rk+1 ) = (1/w)(rk+1 ).
βj
βj
βl
ε2 (1/w)(rk ) =
Thus inf k
(1/w)(rk )
(1/w)(rk+k1 )
≥ ε2 .
It remains to show (L2). As before we choose ε2 = ε0 and k2 = k0 . We
have to prove that there is N0 ∈ N such that for every k ≥ N0 ,
(1/w)(rk+k2 ) < (1 − ε2 )(1/w)(rk ).
273
There is k1 ∈ N such that for every k ≥ k1 and for every n ∈ N we have
wn (rk+k2 ) < (1 − ε2 )wn (rk ).
We fix k ≥ k1 and select 0 < k + k2 < s < 1. Hence, by (6),
1
(1/w)(z) = max
wn (z)
1≤n≤k(s) βn
for every z ∈ D with |z| ≤ s. We put N0 := k1 and distinguish the following
cases:
(i) If (1/w)(rk+k2 ) = (1/βj )wj (rk+k2 ) and (1/w)(rk ) = wj (rk ), then
(1/w)(rk+k2 ) =
1
1
wj (rk+k2 ) ≤ (1 − ε2 )
wj (rk ) = (1 − ε2 )(1/w)(rk ).
βj
βj
(ii) If (1/w)(rk+k2 ) = (1/βj )wj (rk ) and (1/w)(rk ) = (1/βl )wl (rk ), then
1
1
(1/w)(rk+k2 ) =
wj (rk+k2 ) ≤ (1 − ε2 )
wj (rk )
βj
βj
1
≤ (1 − ε2 ) wl (rk ) = (1 − ε2 )(1/w)(rk ).
βl
The claim follows.
The following is a special case of Bierstedt [5, Satz 3.5(1)].
Theorem 26 (Bierstedt [5, Satz 3.5(1)]). Let U1 = (u1,k )k∈N , U2 =
(u2,k )k∈N , . . . , Um = (um,k )k∈N be increasing sequences of strictly positive
continuous and radial functions on D. If
m
m
nO
o
O
W :=
Ui =
ui,k ; k ∈ N ,
i=1
i=1
then
HW0 (Dm ) =
O
f m
ε i=1
(HUi )0 (D).
Since the tensor product of two Fréchet spaces with (QNo) also satisfies
(QNo) (see Peris [33]) we get the following consequence.
Corollary 27. Let U1 = (u1,k )k∈N , U2 = (u2,k )k∈N , . . . , Um = (um,k )k∈N
be increasing sequences of strictly positive continuous and radial functions
on D such that each H(Ui )0 (D) contains the polynomials. Then HW0 (Dm ) =
N
(f ε)m
i=1 H(Ui )0 (D) satisfies (QNo) if and only if each H(Ui )0 (D) has (QNo).
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Mathematical Institute
University of Paderborn
D-33095 Paderborn, Germany
E-mail: [email protected]
Received May 10, 2005
Revised version November 22, 2005
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Weighted Fréchet spaces of holomorphic functions

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